We therefore find that, as the Artin conjecture for odd representations has been proven in [1]}, an eigenform of weight 1 and level \(653^2\)  that reduces to the characteristic 2 form of level 653 we found above. We can additionally twist by the nontrivial character of \(\mathbb {Q}(\sqrt{653})/\mathbb {Q}\) , not changing the determinant or level, to get a second Artin representation, and hence a second modular form of the same weight and nebentypus. These two eigenforms are congruent mod 2, so their average is also an integral form, and there is therefore a nilpotent element of the weight 1 mod 2 Hecke algebra, in a similar sense to [2]}. And conjugating the \(\mathbb {F}_4\) -forms, we obtain 2 more weight 1 forms of level 653. So the index of \({\operatorname{an}}\)  in \( must be at least \) 16\(.\) Indeed, we can find the following four (non-eigen)forms of weight 2 and level 653:
\(f_1=0 &q^{1} &+ 1 &q^{2} &+ 2 &q^{3} &- 4 &q^{4} &+ 0 &q^{5} &+ 2 &q^{6} &+ 0 &q^{7} &+ 4 &q^{8} &+ 0 &q^{9} &+ 4 &q^{10} &+ 0 &q^{11} &+ 1 &q^{12} &- 6 &q^{13} &+\ldots \\f_2=0 &q^{1} &+ 0 &q^{2} &+ 2 &q^{3} &- 3 &q^{4} &+ 0 &q^{5} &+ 2 &q^{6} &+ 2 &q^{7} &+ 2 &q^{8} &+ 4 &q^{9} &- 3 &q^{10} &+ 4 &q^{11} &- 6 &q^{12} &+ 0 &q^{13} &+\ldots \\f_3=0 &q^{1} &+ 0 &q^{2} &+ 0 &q^{3} &+ 4 &q^{4} &+ 0 &q^{5} &+ 1 &q^{6} &+ 2 &q^{7} &+ 2 &q^{8} &+ 4 &q^{9} &+ 5 &q^{10} &+ 2 &q^{11} &+ 0 &q^{12} &+ 4 &q^{13} &+\ldots \\f_4=0 &q^{1} &- 2 &q^{2} &- 6 &q^{3} &+ 2 &q^{4} &+ 0 &q^{5} &+ 2 &q^{6} &+ 2 &q^{7} &- 5 &q^{8} &+ 0 &q^{9} &+ 0 &q^{10} &- 2 &q^{11} &- 6 &q^{12} &- 2 &q^{13} &+\ldots \) 
